Can One Maturity Determine a Time-Homogeneous Local Volatility Model?

# Time-independent local volatility


# Time-independent local volatility












Suppose somebody provides us with a surface of European call prices $C(\tau,K)$ where $\tau$ stands for time-to-maturity and $K$ for the strike. By Dupire's results, there is a unique local volatility function $\sigma(\tau,K)$ that generates these prices, and it can be expressed from them as $$ \sigma(\tau,K) = \frac{2C_\tau}{K^2C_{KK}}, $$ here for simplicity I am assuming that interest rate is zero. Now, if we just have $C(T,K)$ for a single maturity $\tau = T$, is that true that there exists a unique time-independent local volatility $\sigma(K)$ that generates this price at that maturity? In case it does, is there an analytic formula for that function?

## Answer by q.t.f. (score 4)

https://quant.stackexchange.com/a/16725

Yes, there is a unique time homogeneous local vol model. This is proven in http://www.sciencedirect.com/science/article/pii/S0304414912002487. There is a slight generalization required that if the option-implied density is zero somewhere, the corresponding local vol is infinite in that region, giving a "gap diffusion".

No, there is no nice formula for the local vol in this case.

## Answer by vanna (score 0)

https://quant.stackexchange.com/a/16673

No.

In practice the local volatility model has a finite number of slices, so a single slice works as well. Now the problem is : how to compute the time derivative ? Well without adding any information you know that $$ C(0,K) = (S_0-K)_+ $$ so you could try $$ C_\tau = \frac{C(\tau,K)-C(0,K)}{\tau} $$ but it is a very crude approximation. What you may want to do instead is first to use the Dupire formula w.r.t. implied volatilities and consider the whole surface flat and equal to you slice, thus taking $\Sigma_\tau=0$. It is equivalent to use the formula w.r.t. total implied variances $w(\tau,K):=\Sigma^2(\tau,K)\tau$ and compute the derivative as above : $$ \lim_{\tau \rightarrow 0_+} w(\tau,K) = 0$$ $$ w_\tau(\tau,K) \eqsim \frac{w(\tau,K)-w(0_+,K)}{\tau} = \Sigma^2(\tau,K)$$

Here is the formula w.r.t. total variances (a good reference is Gatheral's The Volatility Surface: A Practitioner's Guide) :

$$ \sigma^2(\tau,K) = \frac{w_\tau(K)}{\displaystyle1-\frac{y}{w(\tau,K)}w_y(\tau,K)+\frac{1}{4}\left(-\frac{1}{4}-\frac{1}{w(\tau,K)}+\frac{y^2}{w^2(\tau,K)}\right)(w_y(\tau,K))^2+\frac{1}{2}w_{yy}(\tau,K)} $$ where $y=\ln(K/S_O)$ and $w_\tau(K)=\Sigma(K)^2$. You see here that the denominator still depends on $\tau$ through $w$. There is no way around this.

The vanilla price approach will yield the same conclusion since the equivalent time-derivative has to be computed using the Black-Scholes $\theta$ greek (derivative w.r.t. to time to maturity) with the implied volatility at the considered strike. The Black-Scholes theta depends on the time to maturity even without drift.

Also the local volatility generated by an implied volatility surface is unique obviously (satisfies Dupire equation).